16.13 Polygon Area

Contents

Description

Integrate Polygon Area 1.png

The Polygon Area tool calculates a non-self-intersecting polygon in XY plane, using the determinant of a multivariate matrix.

To Use Polygon Area Tool
  1. Highlight the input data in the worksheet or activate the plot that contains the input data.
  2. Select Analysis: Mathematics: Polygon Area from the Origin menu to open the polyarea dialog. The polyarea dialog box uses the polyarea X-Function to compute the area of the polygon. Results are output to the Results Log.

Dialog Options

Input

The input XY range should define an enclosed area.

Area Type
  • Mathematical Area
    The area value can be positive or negative, and is calculated using the formula provided in the Algorithm section.
  • Absolute Area
    The absolute value of the signed area, which is always positive. See the Algorithm section for details.

Examples

  1. Create a new workbook and import the data file <Origin Installation Directory>\Samples\Mathematics\Circle.dat into it.
  2. Highlight Column B and select Plot: Line: Line from the main menu to make a graph.
  3. Make sure that the graph created in the last step is active. Then select Analysis: Mathematics: Polygon Area from the main menu to open the polyarea dialog. Choose Mathematical Area with the Area Type drop-down list. Finally, click the OK button.
    Polyarea example dialog.png
  4. The result is in the Results Log.
    Polyarea example result.png

Algorithm

This X-Function is capable of calculating the (signed) area of a non-self-intersecting polygon in the \(XY\!\) plane. Suppose the vertices of the polygon are \((x_1, y_1), (x_2, y_2), ..., (x_n, y_n)\!\). The mathematical area can be calculated as:

\[Area= \frac{1}{2} \left ( \begin{vmatrix} x_1 & x_2 \\ y_1 & y_2 \end{vmatrix} + \begin{vmatrix} x_2 & x_3 \\ y_2 & y_3 \end{vmatrix} +...+ \begin{vmatrix} x_n & x_1 \\ y_n & y_1 \end{vmatrix} \right )\]

\[= \frac{1}{2} \left ( x_1y_2-x_2y_1+x_2y_3-x_3y_2+...+x_{n-1}y_n-x_ny_{n-1}+x_ny_1-x_1y_n \right )\]

In the case of a convex polygon, if the vertices are listed sequentially in a counterclockwise direction, the mathematical area of the polygon will be positive; if listed in a clockwise direction, the mathematical area will be negative.

The absolute area of the polygon is calculated as the absolute value of the mathematical area.